Math and Science Discover the Unseen Planets of Trappist-1 – Now That’s Cool!

I am sure by now you have heard about NASA’s discovery of 7 – earth-like planets orbiting the star called Trappist-1 by using the Spitzer Space Telescope. And, apparently 3 of them could possibly be habitable for life. All of this is amazing in itself, but, what is even more amazing is they discovered these planets without really even seeing them.

What?!  How is this possible? How do they know then that there are even planets if they can’t see them? It all comes down to some amazing technology, some data collection, a lot of math, science and analysis. If you are looking for ways to get your students excited about math and science and real-world applications to answer questions, you need look no further.

While listening to a story on NPR, as usual, an astronomer came on to discuss how these planets were in fact discovered. In his discussion, I was just floored by all the applications of geometry and statistics used in this discovery. When he said they couldn’t actually ‘see’ any of the planets, but instead, used the dimming light of the star, Trappist-1, that these planets orbit around as an indication that there were in facts objects/planets orbiting about the star. So – basically,  looking at the stars brightness from the Earth, the amount of starlight that is blocked as each planet passes across the view of the star was used to calculate the size of each planet.  Based on the amount of dimming, they were able to determine the size of the planets, relative to Earth, with the dips in the stars light indicating how fast the individual planets were orbitting the star. This video below explains the process really well:

The star, Trappist-1, is what they call an ultra-cool dwarf star, which is about the size of Jupiter and significantly cooler than our own sun, and is only 39 light years away from us.  That seems far to me, but apparently in ‘space units’ that’s really close! (Here’s a great problem for students – how many miles would 39 light years represent?) Each planets mass was determined by the amount of tug of each planet on the other. Then, using the size and mass calculations, they estimated each planets density, which then allowed them to extrapolate that six of the planets are probably rocky. Another really interesting thing about all the planets is they appear to be tidally locked, which means the same side of the planet always faces its sun, so one half of the planet is always dark, the other always light. This is based on the length of each planets day, or its spin on its axis (determined by watching the planets for a period of days and seeing how often they crossed the star). The shortest day (compared to an earth day) is 1.5 days, the longest is about 20 days (they still have to collect more data for this last one). I found this great chart on the NASA Jet Propulsion site that compares each of the seven planets (with an artist’s rendering of what they might look like…remember, no one can actually ‘see’ these planets yet)

This infographic displays some artist's illustrations of how the seven planets orbiting TRAPPIST-1 might appear — including the possible presence of water oceans — alongside some images of the rocky planets in our Solar System. Information about the size and orbital periods of all the planets is also provided for comparison; the TRAPPIST-1 planets are all approximately Earth-sized.

This infographic displays some artist’s illustrations of how the seven planets orbiting TRAPPIST-1 might appear — including the possible presence of water oceans — alongside some images of the rocky planets in our Solar System. Information about the size and orbital periods of all the planets is also provided for comparison; the TRAPPIST-1 planets are all approximately Earth-sized.

I find the whole process exciting, interesting, and fascinating. I think students would too and there is so much application of mathematics and science going on here. And, as a certified sci-fi geek, just thinking of the possibilities of other life on those ‘M’ class planets (shout out to my fellow Star Trek groupies) is sparking my imagination. Right now, we don’t have the technology to see these seven planets, but who knows? Maybe a student who explores the math and science behind these now might create that next telescope that lets us see the planets, or the space ship that allows us to travel there? Fun to imagine, and fun for students to explore these ‘brave new worlds where no man has gone before….”.

If you are interested in finding out more about this Trappist-1 discovery, here are some more links:

  1. http://www.vox.com/2017/2/22/14698030/nasa-seven-exoplanet-discovery-trappist-1
  2. http://www.csmonitor.com/Science/Spacebound/2017/0222/Exoplanet-update-Discovery-of-seven-Earth-like-planets-heats-up-search-for-life-video
  3. https://www.washingtonpost.com/news/speaking-of-science/wp/2017/02/22/scientists-discover-seven-earthlike-planets-orbiting-a-nearby-star/?utm_term=.801bc7e5c159
  4. http://www.foxnews.com/science/2017/02/23/keys-to-life-scientists-explain-how-newly-discovered-exoplanets-could-be-habitable.html
  5. https://www.sciencenewsforstudents.org/article/new-solar-system-found-have-7-earth-size-planets
  6. http://www.spitzer.caltech.edu/images/6286-ssc2017-01f-TRAPPIST-1-Statistics-Table
  7. https://exoplanets.nasa.gov/news/1419/nasa-telescope-reveals-largest-batch-of-earth-size-habitable-zone-planets-around-single-star/

A Math Nerd’s Dream Museum

img_3760I went to the National Museum of Mathematics (MoMath) today – what else would I do while in NYC?!!  If you were unaware, this is yet another img_3766great attraction to add to your to-do list next time you are in New York City. I was lucky enough to have a few hours today to myself and thoroughly enjoyed my hands-on experiences – me and several hundred school-age children.

The museum is focused on providing hands-on, interactive img_3782mathematical experiences so students can see, create, and play with mathematics.  There are games, art exhibits, bikes with square wheels to ride, cars to control around a mobius strip, img_3780angles, tessellations, fighting robots, logic puzzles….it was really fun, and there was a lot of ‘learning’ embeddedimg_3775 in all of the exhibits, though I did find I was the only one reading – the kids wanted to just ‘do’. But can you really blame them?

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One of my favorite exhibits when you walk into the museum is the wall of etchings done on metal plates. There are parabolic lights above them that move and due to the angles the metal etchings are at, it appears the whole display is moving and that the etches are 3D when in fact they are flat. The etchings themselves are beautiful – lots of mobius strips in there!!  I tried to capture it on video but it doesn’t do it justice.

Another favorite was the art exhibit showing the amazing geometric img_3770sculptures of Miguel Berrocal – famous for creating sculpture puzzles – i.e. sculptures built by pieces fitting together. There are numerous sculptures on display along with puzzle books showing the steps to build some of the sculptures. There are also two hands-on opportunities to try to build some of the sculptures. I tried my hand at the above sculpture, “portrait de Michele”, which they recreated the pieces using a 3D printer and then provide ‘directions’ to build.  My results are below….I was very proud of myself!

There was a little bit of everything – I made myself into a human fractal tree (that’s me as the trunk if you look really close). And then I made two 3D shapes (sphere and star) by putting together flat plates with 2D shapes (circles and triangles) in a layered order so that they end up looking 3D.  That was a challenge trying to piece the different sized shapes in the right order.

There was a lot more fun to be had – from the square tire bike to the shape challenges to building polyhedra. All in all, a fun-filled few hours doing some math and experiencing students enjoying doing math as well. If you ever get the chance to get to NYC, be sure to include the MoMath in your itinerary!

Numeracy – Skills for Life

I just watched this very interesting, and slightly alarming, TedX talk by Alan Smith. It drew my interest because of the title: Why You Should Love Statistics. Statistics is one of those math topics that I really believe all students in high school should take, yet it is often considered secondary in importance to pre-calculus or Algebra 2. My feelings about Statistics is that it is more important for the majority of students (and adults) because statistics are used daily and without an understanding of statistics, it is possible to be continuously deceived or misled. I think our present day political climate is a clear indication of this.  It all comes back to numeracy and understanding numbers and what those numbers, or data, are telling us about the world around us.

Smith begins his talk with some information about numeracy in the UK and then shows some data from the OECD Survey of Adult Skills (PIAACX2012) comparing the numeracy rates from 12 countries, shown below.

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There is a clear problem numeracy in several countries.

Smith goes on to talk about statistics and how statistics are about us – “the science of dealing with data about the state, or the community that we live in….it’s about us as a group, not us as individuals”. He goes on to show how the way people perceive statistics is remarkably different than the reality of those statistics. It’s much more interesting to actually listen and watch Smith as he talks and shows results, so here’s the talk:

His basic message is we need to be more excited about numbers and statistics in general because it gives information about us. And if we are excited about numbers, we become more engaged in looking at numbers and data which can only improve our understanding of numeracy rates and more importantly, an understanding of us. Something much needed in this day and age.

Global Warming? Let’s Look at Some Data

I realize that I am most likely among the minority of folks when I say I miss snow. I have lived in the Philadelphia area going on 3 1/2 years now, and this ‘winter’ has to be one of the most disappointing ones so far.  I think we’ve seen maybe 3 days of snow – less than 3 inches, and all gone in a couple hours.  I haven’t even had to shovel or scrape the car but one time…. There has been a lot of rain. It’s raining today, and suppose to get to 60. Yep – sounds like spring to me, NOT winter! Where’s my snow? Where’s the sledding?

I grew up in Virginia and spent most of my life in Virginia, where we got a lot of snow – I remember some pretty amazing snow storms and tobogganing down the driveway with my brothers and sister. I then moved to Houston, TX for five years back in 2008 and basically lost any hope of seeing snow or even seasons. There is no real winter….no real spring…definitely no change of seasons in Houston, though it is definitely as hot as people say. When we moved back east to the Philly area 3 1/2 years ago, I was so excited to experience a fall again, and my first winter here we had so much snow, we were actually tunneling our way out.  It was great! Sledding at the castle, power outages forcing us to hunker down at the local bars – snowstorms were fun – even the shoveling brought out the neighborhood and a lot of goodwill!

 

The lack of snow this year, and the weird warm temperatures this winter, where it has felt more like spring than winter, has me thinking about whether this is a normal pattern for the area or is it ‘global warming'(which according to our illustrious leader is a hoax), or is it something else? I think it would be an interesting and relevant real-world investigation for students to look at and analyze and make some conclusions and even some predictions, no matter where they live. My guess is lots of you are experiencing some weird weather patterns this ‘winter’ – i.e. Utah & California for example.  I know the kids around here are disappointed there have been no snow days, so they’d probably love the chance to study the numbers and see if this is an expected pattern and hopefully find a chance of snow still exists.

No matter where you live, weather patterns are a great way to analyze data and apply mathematical concepts. Most countries, states, cities and town keep a historical record of weather data – by year, by month, by day.  There are lots of different measures taken into account – temperature (lows & highs), precipitation (rain and snow), barometer pressure, wind, etc. This data is relatively easy to find as well just by doing a simple internet search. Many sites provide customization, where you can specify month, year and other data that you are interested in looking at. I did a relatively simple search for Philadelphia historical data, and compared the month of January from 2013 to 2017 – here are the numbers:

Granted, a little hard to see, but just in a quick glance, students might note that this past January 2017 we had about 5.59 inches of snow fall compared to 19.41 inches in 2016 (all in one day?!!), 3.9 in 2015, 25.86 in 2014, and 3.75 in 2013. Based on this, maybe it’s every other year that we get a lot of snow? Maybe this has nothing to do with global warming? Is there enough data to make these conclusions? Should we be looking at more months or more years? What about the average high or the average lows for each month? Does that make a difference? There are so many interesting questions and comparisons that students could explore with weather data. As a teacher, you could be applying a lot of things like ratio, proportion, measures of central tendency, different types graphical displays, fractions, decimals, algebra.  It’s a font of real-world data that could be used in so many different ways and in so many different math courses. And students would be interested, especially if you are using data from where they live.  Maybe compare the data to other similar cities or other very dissimilar cities. Do a cross-curriculum investigation – i.e. science, language arts, history.

Depending where you live, you can use weather to help students relate mathematics to their own world and explore their environment while doing math. In CA, as an example, you’ve received a tremendous amount of rain this winter – is it enough to end the drought? How long would that take and how much rain? Interesting and relevant questions students would love to investigate. In Utah, how has all the snow impacted the skiing and tourist dollars coming into the state? In Louisiana, South Carolina, Georgia, Florida – how common are tornadoes in ‘winter’?

Lot’s of questions. Lot’s of data out there ready to explore.

One last question – will there be a big snow storm in the Philly area in the next few weeks? I hope the answer is yes…I need a snow day!

Frozen Math

I saw this really cool GIF on FB the other day, showing a bubble freezing. As I watched it, you could see all these beautiful shapes emerging and eventually covering the whole bubble. (I of course wished that it was cold enough where I live for me to go out and try it myself, but alas….where I live seems to be having a no-snow winter this year.

Watch and see:

It looks like snowflakes appearing on the bubble, and snowflakes are fascinating. They are unique, they have amazing patterns that form naturally. Wouldn’t it be fun to explore snowflakes with students? Especially if you live in colder climates where there is actual snow to collect and study. How could we connect the beautiful patterns and unique qualities of snowflakes to mathematics? I set out to explore and found a few great resources for those of you who are interested in exploring frozen math. Yet another way to bring the real-world into the classroom and help students see the math that exists around them.  Even if you don’t live where snow may be, some of these resources provide some great tools for ‘creating’ snowflakes with students.

Here are some links:

  1. http://www.educationworld.com/a_curr/mathchat/mathchat015.shtml This is a nice site because it has several suggestions – from collecting real snowflakes to creating your own, to analyzing patterns and categorizing snowflakes. Great hands-on activities.
  2.  A wide variety of ‘frozen math’ activities here: http://mathwire.com/seasonal/winter05.html including the Koch Curve/Snowflake, where students experience the iterative process to create a snowflake fractal.
  3. Some nice examples and how-to-make paper snowflakes: http://mathcraft.wonderhowto.com/how-to/make-6-sided-kirigami-snowflakes-0131796/
  4. Some nice geometry connections and more paper-snowflake making here: http://playfullearning.net/2015/02/snowflake-math/
  5. This is a great math/science connection with a lot of further embedded links included: http://beyondpenguins.ehe.osu.edu/teaching-about-snowflakes-a-flurry-of-ideas-for-science-and-math-integration
  6. Vi Hart and Doodling is always fun to watch, and here she is doodling and folding with symmetry and fractions: https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart/vi-cool-stuff/v/snowflakes-starflakes-and-swirlflakes

I am sure there are more options out there – these are just a few I stumbled upon in my searching. Don’t let the winter blues set in – get out there and collect some snowflakes and do some frozen math!

Fractions with a Calculator – Looking for Patterns

calculatorI have been working with teachers and using manipulatives, both physical and virtual, to help students think about fractions and develop conceptual understanding about fractional operations, versus just memorizing rules or tricks, as we so often do with students. There are fraction circles or fraction strips that work well as physical manipulatives, and there are several virtual manipulatives as well (i.e. DynamicNumber.org for any Sketchpad users out there, and the National Library of Virtual Manipulatives to give just a couple resources).

Manipulatives are a valuable resource in math class as they allow students to visually represent numbers, manipulate them, get hands-on with the math, and make some connections before moving into just the numerical representation alone. When working with fraction manipulatives, from my own experiences and those I have had with students, the manipulatives can constrain the number of possible examples we can provide students (either because a teacher might not physically have enough for all students or the manipulatives themselves only go up to certain values). As an example, most physical fraction circle manipulatives allow you to work with a limited range of fractional values – halves, thirds, fourths, fifths, sixths, eighths, tenths and twelfths. Virtual manipulatives offer more options, which is nice because students should see more than just common fractional pieces or ‘nice’ fractions – sevenths, or elevenths or twenty-fifths as an example. Obviously, the idea of manipulatives is to provide that hands-on experience, visually see what’s happening, and then create conjectures.

Another tool that is often overlooked, particularly at the elementary level, is the calculator. Obviously, when dealing with fractions, you want a calculator that uses natural display, showing fractions in their numerator over denominator form so students recognize the fractional number. I realize many of you might be thinking that the calculator is a bad choice because it provides the answers….but that in fact is an advantage here when trying to help students recognize patterns and develop their own understanding of fractional operations.  We want students to recognize what seems to be happening – test it out on many examples before they come to a conclusion.  A calculator (like the fx-55Plus shown above) is a great way to do this.  If you don’t have manipulatives, you can actually use a calculator like the fx-55Plus to help students understand fractional operations.

Let’s take fraction addition. Obviously, we are going to start with adding fractions with like denominators.  You can put several different problems into the calculator and students can observe both the added fractions and the answers. Students can talk and share what they notice about the multitude of fractions they are adding (all with like denominators). They can make up their own addition problems and see if the pattern or things they notice hold true. Fraction and answers showing up quickly help them discern patterns because they can quickly see many examples, and use ‘funky’ fractions, not just the typical ones we tend to always rely on (i.e. halves, thirds, etc.). It’s even okay that the numerator might occasionally end up larger than the denominator – the pattern still holds true (i.e. the denominator remains the same, the numerators are added together).

With a calculator, you can use messy fractions with not your typical denominators and even numerators larger than the denominator. For addition, our focus is on what patterns do the students see with the numerator and denominator and do those patterns hold true no matter what fractions we are adding? We can get into simplifying the answers at some point, but at first, the focus is on the addition.

Once students have the idea that with a like denominator, you add the numerators, you can then switch it up. Let’s add fractions with unlike denominators.  You can encourage smaller numbers in the denominator and numerator to start, and then once students think they have the pattern, they can ‘test it out’ with some larger digits in the numerator and denominator. The thing here is the denominators are different and so how does the end result differ (if does) from when the denominators are the same? What might be happening? Test it out.

The beauty of the calculator (again, one like the fx-55plus that quickly and easily shows fractions in their natural display), is that students can create many examples to look for patterns and then quickly test their conjectures on different problems to see if it works. You are encouraging critical thinking, problem solving, and communication using a simple tool that provides much more diverse fraction examples than you can provide with manipulatives alone.

My point – when helping students develop number sense, especially with fractions, don’t rule the calculator out as a tool. You should use multiple tools with students to provide them with different ways to develop their own conceptual understanding. Calculators can be a tool, even at the elementary level.

 

 

 

The Language of Math – Consistency to Support Students

I’ve been teaching some courses at Drexel University, and in those course we really focus on the language of mathematics and using students prior knowledge to help them make connections and build on their mathematical understanding.

In a current course, we are exploring integer addition and using manipulative’s to provide both a visual and concrete connection to the idea of creating zero pairs, and then progressing to the more abstract addition of integers without manipulatives and how do you support students understanding and language. What has come up frequently is the terms ‘cancelling out’ and ‘disappearing’ and ‘opposites’ to explain or help students understand that creating zero pairs allows you to use the additive identity property.  But – what’s really happening, mathematically, is that we are using additive inverse (opposites) to create these zero pairs, which are NOT cancelling out or disappearing, but instead, are creating the quantity zero. And, once we have created this quantity zero, the remaining value can be added to those zeros using the additive identity property. Cancelling out or disappearing implies they don’t exist, which, when we expand the idea identity into multiplication where we use the multiplicative inverse to create a 1 (not a zero), cancelling out really seems confusing.

Seems like it shouldn’t make a difference, but think about it – if we use terms like opposites, cancel out, or disappear in one grade, and then the next grade or future courses, the same ideas are referred to as additive inverse, zero pairs, additive identity property, students will be confused and think they have never seen these concepts before. While it is very important to use language students understand to start the process, I think as mathematics teachers, no matter what grade level, we need to model proper mathematical language in conjunction with the ‘student-friendly’ terms we tend to rely on or fall back on. This way, students can relate, but then learn, build on, connect & utilize mathematical language so that we are all communicating on an equal playing field.

Something as simple as -3 being referred to as negative 3, minus 3, or even 3 negatives. That’s confusing. Let’s learn to be consistent with our mathematical language and use the correct vocabulary – both in our own teaching, but also in what we expect to hear from our students. So if a student says that the opposite of 3 is -3 (positive 3 and negative 3), let’s acknowledge that they are thinking correctly, and in mathematics we refer to that as an inverse. This will then help them make that connection when we talk about inverse operations.  Consistent mathematical language supports students understanding as they progress into more abstract mathematics.

We want our students to communicate mathematically with the language of mathematics and become proficient mathematicians. Let’s then make a conscientious effort to use and model correct mathematical language instead of the ‘short-cuts’ or ‘simplified language’ we tend to use.  Again – important to start out with this type of language to help students connect prior knowledge, but then more important to model using and support students use of, correct mathematical language. I think it would go a long way in preventing some confusion students experience as they move from grade to grade or course to course.